Continuum limit of the lattice Lohe group model and emergent dynamics

Abstract

We study the emergent dynamics and global well‐posedness of the matrix‐valued integro‐differential equation which can be derived from the continuum limit of the lattice Lohe group model. The lattice Lohe group model corresponds to the generalized high‐dimensional Kuramoto model. The solution to the lattice Lohe group model can be cast as a simple function‐valued solution to the continuum Lohe group model. We first construct a local classical solution to the continuum Lohe group model, and then we find an invariant set and derive a global well‐posedness in some sufficient frameworks formulated in terms of initial data, system functions, and system parameters. We also show that phase‐locked states can emerge from the admissible class of initial data in a large coupling regime. Moreover, we show that sequence of simple functions obtained from the solutions of the lattice Lohe group model converges to a local classical solution to the continuum Lohe group model in supremum norm.